The k-th shape homotopy group of a space, defined as the inverse limit of the homotopy groups of an HPol-expansion, carries no topology of its own; this Idea covers the natural topology placed on it so that it becomes a Hausdorff topological group for every topological space. Three descriptions of the topology coincide: the inverse limit topology inherited from the discrete homotopy groups of the expansion's polyhedra, the topology induced from the quasitopological homotopy group of the shape-loop space, and the quotient topology on shape loop classes. Consequences include preservation under finite products, discreteness exactly when the inverse system is stable, and the fact that two spaces can share all shape homotopy groups as abstract groups yet be separated by them as topological groups.
Shape theory replaces a topological space by an inverse system of polyhedra (an HPol-expansion), so that spaces too badly behaved for homotopy theory still carry invariants; the kth shape group of a …